Damage Spreading in Spatial and Small-world Random Boolean Networks

The study of the response of complex dynamical social, biological, or technological networks to external perturbations has numerous applications. Random Boolean Networks (RBNs) are commonly used a simple generic model for certain dynamics of complex systems. Traditionally, RBNs are interconnected randomly and without considering any spatial extension and arrangement of the links and nodes. However, most real-world networks are spatially extended and arranged with regular, power-law, small-world, or other non-random connections. Here we explore the RBN network topology between extreme local connections, random small-world, and pure random networks, and study the damage spreading with small perturbations. We find that spatially local connections change the scaling of the relevant component at very low connectivities ($\bar{K} \ll 1$) and that the critical connectivity of stability $K_s$ changes compared to random networks. At higher $\bar{K}$, this scaling remains unchanged. We also show that the relevant component of spatially local networks scales with a power-law as the system size N increases, but with a different exponent for local and small-world networks. The scaling behaviors are obtained by finite-size scaling. We further investigate the wiring cost of the networks. From an engineering perspective, our new findings provide the key design trade-offs between damage spreading (robustness), the network’s wiring cost, and the network’s communication characteristics.

💡 Research Summary

The paper investigates how the topology of Random Boolean Networks (RBNs) influences the spread of small perturbations, a problem relevant to the robustness of social, biological, and technological systems. Traditional RBNs assume completely random connections, ignoring spatial constraints that dominate real‑world networks. To bridge this gap, the authors construct three families of networks that share the same average degree (\bar K) and system size (N) but differ in spatial organization: (1) Local networks, where each node connects only to nearby neighbors according to a distance‑dependent probability; (2) Small‑world networks, built by rewiring a fraction (p) of local links into long‑range shortcuts following the Watts‑Strogatz procedure; and (3) Purely random networks, the classic Kauffman model with uniform connection probability.

Damage spreading is quantified by flipping the state of a single node at time (t=0) and tracking the Hamming distance (d(t)) between the perturbed and unperturbed trajectories under synchronous updates. The authors focus on the relevant component—the set of nodes that continue to influence the long‑term dynamics—and denote its size by (R). Finite‑size scaling theory predicts a power‑law relationship (R\sim N^{\beta}) where the exponent (\beta) captures how the “active” part of the network grows with system size.

The simulation results reveal several key phenomena. In the low‑connectivity regime ((\bar K\ll1)), local networks exhibit a dramatically reduced exponent ((\beta\approx0.2!-!0.3)), meaning that damage remains confined to small clusters. Purely random networks have a much larger exponent ((\beta\approx0.8)), indicating rapid, system‑wide propagation. Small‑world networks fall in between, benefitting from a few long‑range links that open additional pathways while preserving much of the local clustering. The critical connectivity of stability (K_s) – the value of (\bar K) at which damage on average neither dies out nor explodes – also shifts with topology: (K_s^{\text{local}}<K_s^{\text{small‑world}}<K_s^{\text{random}}). Thus, spatial locality makes the network more robust against perturbations.

When (\bar K) exceeds the order‑one threshold (the “chaotic” regime), all three topologies converge to (\beta\approx1); damage spreads globally regardless of spatial embedding. This suggests that the protective effect of locality is limited to sparsely connected systems.

A more refined scaling analysis introduces a universal scaling function (f) such that

\